Burnside Problem: Is a finitely generated periodic group of bounded exponent necessarily finite?

Definition
Let Fm denote the free group of rank m. For a fixed n let Fmn denote the subgroup of Fm generated by gn for g ∈ G.
Then Fmn is a normal subgroup of Fm(it is even an invariant subgroup), and we define the Burnside GroupB(m, n) to be the factor group Fm/ Fmn .

These results imply that any counterexample to the Burnside Problems will have to be difficult, i.e. not expressible in terms of the well-known linear groups. After this initial flurry of results, no more progress was made on the Problems until the early 1930's, when the topic was resurrected by the suggestion of a variant on the original problem:

Restricted Burnside Problem: Are there only finitely many finite m-generator groups of exponent n?

If the Restricted Burnside Problem has a positive solution for some m, n then we may factor B(m, n) by the intersection of all subgroups of finite index to obtain B0(m,n), the universal finite m-generator group of exponent n having all other finite m-generator groups of exponent n as homomorphic images.

Note that if B(m,n) is finite then B0(m,n) = B(m,n).

Despite this formulation having been present on the seminar circuit in the 1930's, it was not until 1940 that the first paper, by Grün [6], appeared specifically addressing the RBP, and not until 1950 that the term "Restricted Burnside Problem" was coined by Magnus [7].

Now (moving ahead), the classification of finite simple groups in the 1980's shows that ii. and iii. hold. Even earlier it was known for n odd by Feit-Thompson (the "odd-order paper" of 1962), and at the time of publication must have been a reasonable conjecture.

Consequently, to prove that B0(m,n) exists for all m, n we need only (!) show that B0(m, pk) exists for all m and prime powers pk. Kostrikin had "shown" that B0(m, p) exists.

In 1989 Zelmanov announced his proof of a positive solution of the Restricted Burnside Problem and was awarded a Fields medal for this in 1994.

1959

Turning back to the original Burnside Problems, Novikov announced that B(m, n) is infinite for n odd, n > 71. Novikov published a collection of ideas and theorems [14], but no definitive proof was forthcoming. John Britton suspected Novikov's proof was wrong and he began to work on the problem.

1964

Golod and Shafarevich [15] provided a counter-example to the General Burnside Problem -- an infinite, finitely generated, periodic group.

This saddened Britton since he was close to publishing himself, but he continued and finished in 1970. His paper was published in 1973, but Adian discovered that it was wrong. There was not a single error in any lemma. However in order to apply them simultaneously the inequalities needed to make their hypotheses valid were inconsistent. Britton never really recovered, and this was to be the last major research paper he published.